For certain values of do it.

Doctors would never say things like "everyone should learn medicine" or "anyone can be a doctor." Instead they erect barriers to entry to eliminate competition and force you to pay for their services, driving up the cost of health care, and the stupid general public loves them for it.

Doctors, like real Architects (i.e. not software or information ones!), and real Engineers, are a regulated and certified profession because when they make mistakes, someone dies, not because it is an economic advantage for them to be so. What you are describing is something more akin to the guilds system for craftsmen, which is not something I think you should aspire to or want to return to.

I imagine some part of software production will become codified and regulated in the same way in time - that part which deals with life or death situations or produces medical equipment for example. As software grows to eat the world this might start to encompass more of the profession, but probably there will always be a large area of software which just doesn't matter enough to be codified and regulated (I'd include in that most of the work that companies like Apple, Amazon, Google, and web startups do - advertising or most websites can lose money when they go down, but it doesn't risk lives). I'm sure you can think of a case where software risks lives, but the vast majority does not. So I don't think this comparison is helpful.

As to driving up the cost of healthcare, try comparing countries and you'll see costs are more related to the system of insurance and breadth of coverage than how much doctors are paid.

What is wrong with programmers? Are we simply too egalitarian for our own good?

Perhaps it is enlightened self-interest. The more people who know how useful software can be (if only in a small way), the more willing they will be to commission it. Monopolies of knowledge are easily undermined anyway, and why not try to improve the lives of others in a small way by encouraging them to automate filling in their spreadsheet or other daily tasks with a little scripting for example?

If you have sufficient experience or talent, lots of other people learning to code/program/engineer/whatever label you prefer should be the last of your concerns.